|
titre non précisé| old_uid | 18391 |
|---|
| title | titre non précisé |
|---|
| start_date | 2020/10/09 |
|---|
| schedule | 11h30 |
|---|
| online | no |
|---|
| summary | We consider a model of (biological) neurons in interaction.
Each neuron is characterized by its membrane potential, assumed to be of
``Integrate-And-Fire’’ type : between two successive spikes, the membrane
potential (V_t) solves an ODE. The neuron spikes at rate f(V_t)
(it only depends on the membrane potential of this neuron). At the
spiking time, the membrane potential is reset to a resting value.
At this same time, the discharge is propagated to the other neurons
of the network through a jump in the membrane potential. Altogether the finite system with N neurons is a Piecewise Deterministic Markov Process.
We are interested here in the asymptotic behavior as the number of neurons goes to
infinity : a typical neuron in the limit system follows a McKean-Vlasov SDE.
We study it (existence/uniqueness and invariant distributions).
Furthermore, we prove that the local stability of a given invariant distribution can be characterized through the location of the roots of an
explicit holomorphic function.
We finally discuss the existence of periodic solutions through a Hopf bifurction.
An important tool is the Volterra integral equation associated to the process. |
|---|
| responsibles | Bachir |
|---|
| |
|